On a generalized doubly parabolic Keller-Segel system in one spatial dimension

TL;DR

This paper investigates a fractional Keller-Segel system in one dimension, establishing well-posedness, analyzing chaotic dynamics with peaks, and exploring finite-time blow-up scenarios through numerical simulations.

Contribution

It generalizes existing results to fractional diffusions, providing new insights into the system's dynamical behavior and blow-up phenomena.

Findings

Existence of local and global solutions under certain conditions

Chaotic spatio-temporal behavior with peak formation

Numerical evidence of finite-time blow-up with weak diffusion

Abstract

We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic-elliptic fractional case.